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First of all note that from our first derivation of the set inclusion it is clear that ⎛ ⎞ ⎝ ∂x f(x0 , y)⎠ ⊆ ∂ϕ(x0 ). y∈Yˆ (x0 ) This shows that the set ⎛ ⎞ ⎝ ∂x f(x0 , y)⎠ y∈Yˆ (x0 ) is bounded since ∂ϕ(x0 ) is a compact and convex set. We shall now have to show that the set on the left-hand side of the above inclusion is closed. Let ξk ∈ ∂x f(x0 , y k ), with y k ∈ Yˆ (x0 ), be a convergent sequence and let us assume that ξk → ξ∗ . Since Yˆ (x0 ) is a compact set, we conclude without loss of generality that y k → y ∗ ∈ Yˆ (x0 ).